Question: Line $m$ has the equation $y = 3x + 5$.  Line $n$ has the equation  $y = kx - 7$.  Lines $m$ and $n$ intersect at the point $(-4, -7)$.  What is the value of $k$?
Since both lines intersect at the point $(-4,-7)$, line $n$ must pass through this point.  We can substitute these coordinates into the equation $y=kx-7$ and solve for $k$ as shown: \begin{align*}
-7&=k(-4)-7\\
\Rightarrow\qquad -7&=-4k-7\\
\Rightarrow\qquad 0&=-4k\\
\Rightarrow\qquad \boxed{0}&=k
\end{align*}